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Chapter 14: Problem 13

Find the divergence of the following vector fields. $$\mathbf{F}=\left\langle x^{2}-y^{2}, y^{2}-z^{2}, z^{2}-x^{2}\right\rangle$$

Short Answer

Expert verified
Answer: The divergence of the given vector field is \(2x + 2y + 2z\).

Step by step solution

01

Identify the components

The given vector field is \(\textbf{F} = \langle x^2 - y^2, y^2 - z^2, z^2 - x^2 \rangle\). We can see that the components of the vector field are: $$F_x = x^2 - y^2$$ $$F_y = y^2 - z^2$$ $$F_z = z^2 - x^2$$
02

Compute the partial derivatives

To find the divergence, we need to compute the partial derivatives of each component with respect to its corresponding coordinate: $$\frac{\partial F_x}{\partial x} = 2x$$ $$\frac{\partial F_y}{\partial y} = 2y$$ $$\frac{\partial F_z}{\partial z} = 2z$$
03

Calculate the divergence

Now that we have the partial derivatives, we can calculate the divergence of the vector field using the following formula: $$\text{div}\,\textbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$ Substituting the values that we found in Step 2, we get: $$\text{div}\,\textbf{F} = 2x + 2y + 2z$$
04

Final result

The divergence of the given vector field is: $$\text{div}\,\textbf{F} = 2x + 2y + 2z$$

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Most popular questions from this chapter

The potential function for the force field due to a charge \(q\) at the origin is \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{|\mathbf{r}|},\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of a point in the field and \(\varepsilon_{0}\) is the permittivity of free space. a. Compute the force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is \(\nabla \times \mathbf{F}=\mathbf{0}\).

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \cdot(\varphi \mathbf{F})=\nabla \varphi \cdot \mathbf{F}+\varphi \nabla \cdot \mathbf{F} \quad \text { (Product Rule) }$$

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$

Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?

A square plate \(R=\\{(x, y): 0 \leq x \leq 1,\) \(0 \leq y \leq 1\\}\) has a temperature distribution \(T(x, y)=100-50 x-25 y\) a. Sketch two level curves of the temperature in the plate. b. Find the gradient of the temperature \(\nabla T(x, y)\) c. Assume that the flow of heat is given by the vector field \(\mathbf{F}=-\nabla T(x, y) .\) Compute \(\mathbf{F}\) d. Find the outward heat flux across the boundary \(\\{(x, y): x=1,0 \leq y \leq 1\\}\) e. Find the outward heat flux across the boundary \(\\{(x, y): 0 \leq x \leq 1, y=1\\}\)
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